Theorem fun2ssres 3559

Description: Equality of restrictions of a function and a subclass.

Assertion

Ref	Expression
fun2ssres	⊢ ((Fun F ⋀ G ⊆ F ⋀ A ⊆ dom G) → (F ↾ A) = (G ↾ A))

Proof of Theorem fun2ssres

Step	Hyp	Ref	Expression
1		resabs1 3394	. . . 4 ⊢ (A ⊆ dom G → ((F ↾ dom G) ↾ A) = (F ↾ A))
2	1	eqcomd 1483	. . 3 ⊢ (A ⊆ dom G → (F ↾ A) = ((F ↾ dom G) ↾ A))
3		funssres 3558	. . . 4 ⊢ ((Fun F ⋀ G ⊆ F) → (F ↾ dom G) = G)
4		reseq1 3374	. . . 4 ⊢ ((F ↾ dom G) = G → ((F ↾ dom G) ↾ A) = (G ↾ A))
5	3, 4	syl 10	. . 3 ⊢ ((Fun F ⋀ G ⊆ F) → ((F ↾ dom G) ↾ A) = (G ↾ A))
6	2, 5	sylan9eqr 1532	. 2 ⊢ (((Fun F ⋀ G ⊆ F) ⋀ A ⊆ dom G) → (F ↾ A) = (G ↾ A))
7	6	3impa 830	1 ⊢ ((Fun F ⋀ G ⊆ F ⋀ A ⊆ dom G) → (F ↾ A) = (G ↾ A))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ⊆ wss 2050  dom cdm 3176   ↾ cres 3178  Fun wfun 3182

This theorem is referenced by:  tfrlem9 3925  tfrlem11 3927  subgres 8113

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-res 3196  df-fun 3198

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